Derivation of the temperature dependence of the order parameter for solid orthohydrogen and paradeuterium

Abstract

The orientational free energy and order parameter for solid orthohydrogen and paradeuterium are obtained using a Green's-function diagrammatic technique developed by the authors. The orientational interaction is assumed to be electrostatic quadrupole-quadrupole. The zero-phonon, and $J=1\ensuremath{\rightarrow}J=3$ effects on the order parameter are also taken into account. The expressions for the free energy of the orientationally ordered and disordered phase are obtained in $\frac{1}{z}$ approximation, where $z=12$ is the number of the nearest neighbors in the fcc lattice. An equation for the order parameter is obtained from the minimum condition for the free energy. The orientational phase transition is predicted to be first order. The critical temperature is found to be 3.64 K for ortho-${\mathrm{H}}_{2}$ and 4.63 K for para-${\mathrm{D}}_{2}$. The temperature ${T}_{1}$ and ${T}_{2}$ which are the maximum (${T}_{1}$) and minimum (${T}_{2}$) metastability limit for the ordered and disordered phase, respectively, are also calculated. The are ${T}_{1}=0.60$ ${T}_{0}$ and ${T}_{2}=0.46$ ${T}_{0}$, where ${T}_{0}$ is the critical temperature obtained from the molecular-field theory. The results are in a good agreement with experimental data.